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June 19, 2005

Reconnection Probability

Much of the recent resurgence of interest in cosmic strings has to do with the possibility that such strings might be string-theoretic in nature, and that their properties might be observationally-distinguishable from those of “ordinary” field-theoretic cosmic strings.

It has been taken for granted that, whatever their microscopic origins, cosmic strings could be characterized by their string tension, GμG\mu, and their reconnection probability, PP. Observationally, we need Gμ⪅10−7G\mu\lessapprox 10^{-7}.

In the Abelian Higgs model, P=1P=1. The same is true is a surprisingly wide variety of weakly-coupled field theory generalizations. By contrast, P∼gst2P\sim g_{st}^2 for weakly-coupled fundamental strings and P<1P\lt 1for D-strings. For QCD flux strings, P∼1/Nc2P\sim 1/N_c^2.

This difference in the reconnection probability is taken to be one of the hallmarks of string-theoretic cosmic strings. (The other being the fact that (p,q)(p,q)-strings form 3-string junctions.) You might ask to what extent do we know that’s true and, more generally, to what extent do we understand the range of behaviours available in field-theoretic cosmic strings?

One can rather cheaply get P≪1P\ll 1, even in weakly-coupled field theory. Take NN identical decoupled Abelian Higgs models. There are then NN superficially indistiguishable types of strings. Strings of the same type reconnect with probability 11; strings of different types pass right through each other. So, overall, the observable reconnection probability is 1/N1/N.

That’s sorta cheap, but, recently, Hashimoto and Tong considered a more interesting model. Let ϕi\phi_i be NfN_f scalars in the fundamental representation of U(Nc)U(N_c). The Lagrangian
L=14e2Tr(Fμν2)+∑i=1NfDμϕi†Dμϕi−λe22Tr(∑i=1Nfϕiϕi†−v2𝟙)2+Lsoft
L = \frac{1}{4e^2} Tr(F_{\mu\nu}^2) + \sum_{i=1}^{N_f} D_\mu \phi_i^\dagger D^\mu \phi_i - \frac{\lambda e^2}{2} Tr\left(\sum_{i=1}^{N_f} \phi_i\phi_i^\dagger -v^2 &#x1D7D9; \right)^2 + L_{\text{soft}}
where LsoftL_{\text{soft}} explicitly breaks the SU(Nf)SU(N_f) global symmetry of the model. Neglecting LsoftL_{\text{soft}}, the theory is invariant under U(Nc)×SU(Nf)U(N_c)\times SU(N_f). For e2v2≫ΛQCDe^2 v^2 \gg \Lambda_{QCD}, the theory is weakly coupled, and the symmetry is broken spontaneously. For Nc=Nf≡NN_c=N_f\equiv N,
U(Nc)×SU(Nf)→SU(N)diag
U(N_c)\times SU(N_f)\to SU(N)_{\text{diag}}
Since π1\pi_1 of the vacuum manifold is ℤ\mathbb{Z}, the low-energy theory has strings. But, unlike the Abelian Higgs model, the holonomy of the gauge connection around the string (or vortex in 2+1 dimensions) picks out a direction in the gauge group. Given a pair of strings, the relative orientation in the gauge group is physically significant.

According to Hashimoto and Tong, the strings will reconnect (with probability P=1P=1), unless the relative orientation in the gauge group is such that the strings lie in commuting, mutually-orthogonal U(1)U(1) subgroups of the gauge group.

That requires a fine-tuning. So, for generic initial conditions, the strings will reconnect.

Things change when we include the explicit symmetry-breaking term
Lsoft=12Tr(DμΦDμΦ)−∑iϕi†(Φ−mi)2ϕi−12m2Tr(Φ2)
L_{\text{soft}}= \frac{1}{2}Tr(D_\mu \Phi D^\mu\Phi) - \sum_i \phi_i^\dagger (\Phi-m_i)^2 \phi_i - \frac{1}{2}m^2 Tr(\Phi^2)
where Φ\Phi is an adjoint-valued scalar field. Now, instead of a continuous family of strings, there are NN distinct ones, spanning the Cartan subalgebra. They all have the same tension, and so are indistinguishable. Indeed, at sufficiently low energies, it’s just like NN decoupled Abelian Higgs models, and P=1/NP=1/N. But, at energies above the scale of explicit symmetry-breaking (but still low enough so that the moduli space approximation is valid), the behaviour returns to that of the model without LsoftL_{\text{soft}}, and P=1P=1.

So we have an example of a velocity-dependent reconnection probability.

That’s very interesting. It ought to have a rather dramatic impact on the evolution of string networks. All the simulations assume that PP is just a constant, rather than being velocity-dependent. Now we have an effect which makes fast-moving strings reconnect more readily than slow-moving ones.

Posted by distler at June 19, 2005 12:51 AM

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